3.1.44 \(\int \frac {1}{(a \text {sech}^3(x))^{5/2}} \, dx\) [44]

Optimal. Leaf size=121 \[ -\frac {26 i F\left (\left .\frac {i x}{2}\right |2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}} \]

[Out]

-26/77*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))/a^2/cosh(x)^(3/2)/(a*sech(x)^3)^(1
/2)+78/385*cosh(x)*sinh(x)/a^2/(a*sech(x)^3)^(1/2)+26/165*cosh(x)^3*sinh(x)/a^2/(a*sech(x)^3)^(1/2)+2/15*cosh(
x)^5*sinh(x)/a^2/(a*sech(x)^3)^(1/2)+26/77*tanh(x)/a^2/(a*sech(x)^3)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3854, 3856, 2720} \begin {gather*} \frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}-\frac {26 i F\left (\left .\frac {i x}{2}\right |2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {2 \sinh (x) \cosh ^5(x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \sinh (x) \cosh ^3(x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {78 \sinh (x) \cosh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x]^3)^(-5/2),x]

[Out]

(((-26*I)/77)*EllipticF[(I/2)*x, 2])/(a^2*Cosh[x]^(3/2)*Sqrt[a*Sech[x]^3]) + (78*Cosh[x]*Sinh[x])/(385*a^2*Sqr
t[a*Sech[x]^3]) + (26*Cosh[x]^3*Sinh[x])/(165*a^2*Sqrt[a*Sech[x]^3]) + (2*Cosh[x]^5*Sinh[x])/(15*a^2*Sqrt[a*Se
ch[x]^3]) + (26*Tanh[x])/(77*a^2*Sqrt[a*Sech[x]^3])

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx &=\frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\text {sech}^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (13 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {11}{2}}(x)} \, dx}{15 a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (39 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {7}{2}}(x)} \, dx}{55 a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (39 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {3}{2}}(x)} \, dx}{77 a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (13 \text {sech}^{\frac {3}{2}}(x)\right ) \int \sqrt {\text {sech}(x)} \, dx}{77 a^2 \sqrt {a \text {sech}^3(x)}}\\ &=\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {13 \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}\\ &=-\frac {26 i F\left (\left .\frac {i x}{2}\right |2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 63, normalized size = 0.52 \begin {gather*} \frac {\cosh (x) \sqrt {a \text {sech}^3(x)} \left (-24960 i \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )+19122 \sinh (2 x)+4406 \sinh (4 x)+826 \sinh (6 x)+77 \sinh (8 x)\right )}{73920 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x]^3)^(-5/2),x]

[Out]

(Cosh[x]*Sqrt[a*Sech[x]^3]*((-24960*I)*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2] + 19122*Sinh[2*x] + 4406*Sinh[4*x]
+ 826*Sinh[6*x] + 77*Sinh[8*x]))/(73920*a^3)

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Maple [F]
time = 0.84, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \mathrm {sech}\left (x \right )^{3}\right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sech(x)^3)^(5/2),x)

[Out]

int(1/(a*sech(x)^3)^(5/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sech(x)^3)^(-5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.19, size = 718, normalized size = 5.93 \begin {gather*} \frac {49920 \, \sqrt {2} {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right )^{7} \sinh \left (x\right ) + 28 \, \cosh \left (x\right )^{6} \sinh \left (x\right )^{2} + 56 \, \cosh \left (x\right )^{5} \sinh \left (x\right )^{3} + 70 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{4} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sqrt {2} {\left (77 \, \cosh \left (x\right )^{16} + 1232 \, \cosh \left (x\right ) \sinh \left (x\right )^{15} + 77 \, \sinh \left (x\right )^{16} + 14 \, {\left (660 \, \cosh \left (x\right )^{2} + 59\right )} \sinh \left (x\right )^{14} + 826 \, \cosh \left (x\right )^{14} + 196 \, {\left (220 \, \cosh \left (x\right )^{3} + 59 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{13} + 2 \, {\left (70070 \, \cosh \left (x\right )^{4} + 37583 \, \cosh \left (x\right )^{2} + 2203\right )} \sinh \left (x\right )^{12} + 4406 \, \cosh \left (x\right )^{12} + 8 \, {\left (42042 \, \cosh \left (x\right )^{5} + 37583 \, \cosh \left (x\right )^{3} + 6609 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{11} + 2 \, {\left (308308 \, \cosh \left (x\right )^{6} + 413413 \, \cosh \left (x\right )^{4} + 145398 \, \cosh \left (x\right )^{2} + 9561\right )} \sinh \left (x\right )^{10} + 19122 \, \cosh \left (x\right )^{10} + 4 \, {\left (220220 \, \cosh \left (x\right )^{7} + 413413 \, \cosh \left (x\right )^{5} + 242330 \, \cosh \left (x\right )^{3} + 47805 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{9} + 6 \, {\left (165165 \, \cosh \left (x\right )^{8} + 413413 \, \cosh \left (x\right )^{6} + 363495 \, \cosh \left (x\right )^{4} + 143415 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{8} + 16 \, {\left (55055 \, \cosh \left (x\right )^{9} + 177177 \, \cosh \left (x\right )^{7} + 218097 \, \cosh \left (x\right )^{5} + 143415 \, \cosh \left (x\right )^{3}\right )} \sinh \left (x\right )^{7} + 2 \, {\left (308308 \, \cosh \left (x\right )^{10} + 1240239 \, \cosh \left (x\right )^{8} + 2035572 \, \cosh \left (x\right )^{6} + 2007810 \, \cosh \left (x\right )^{4} - 9561\right )} \sinh \left (x\right )^{6} - 19122 \, \cosh \left (x\right )^{6} + 4 \, {\left (84084 \, \cosh \left (x\right )^{11} + 413413 \, \cosh \left (x\right )^{9} + 872388 \, \cosh \left (x\right )^{7} + 1204686 \, \cosh \left (x\right )^{5} - 28683 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (70070 \, \cosh \left (x\right )^{12} + 413413 \, \cosh \left (x\right )^{10} + 1090485 \, \cosh \left (x\right )^{8} + 2007810 \, \cosh \left (x\right )^{6} - 143415 \, \cosh \left (x\right )^{2} - 2203\right )} \sinh \left (x\right )^{4} - 4406 \, \cosh \left (x\right )^{4} + 8 \, {\left (5390 \, \cosh \left (x\right )^{13} + 37583 \, \cosh \left (x\right )^{11} + 121165 \, \cosh \left (x\right )^{9} + 286830 \, \cosh \left (x\right )^{7} - 47805 \, \cosh \left (x\right )^{3} - 2203 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (4620 \, \cosh \left (x\right )^{14} + 37583 \, \cosh \left (x\right )^{12} + 145398 \, \cosh \left (x\right )^{10} + 430245 \, \cosh \left (x\right )^{8} - 143415 \, \cosh \left (x\right )^{4} - 13218 \, \cosh \left (x\right )^{2} - 413\right )} \sinh \left (x\right )^{2} - 826 \, \cosh \left (x\right )^{2} + 4 \, {\left (308 \, \cosh \left (x\right )^{15} + 2891 \, \cosh \left (x\right )^{13} + 13218 \, \cosh \left (x\right )^{11} + 47805 \, \cosh \left (x\right )^{9} - 28683 \, \cosh \left (x\right )^{5} - 4406 \, \cosh \left (x\right )^{3} - 413 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 77\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{147840 \, {\left (a^{3} \cosh \left (x\right )^{8} + 8 \, a^{3} \cosh \left (x\right )^{7} \sinh \left (x\right ) + 28 \, a^{3} \cosh \left (x\right )^{6} \sinh \left (x\right )^{2} + 56 \, a^{3} \cosh \left (x\right )^{5} \sinh \left (x\right )^{3} + 70 \, a^{3} \cosh \left (x\right )^{4} \sinh \left (x\right )^{4} + 56 \, a^{3} \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, a^{3} \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{7} + a^{3} \sinh \left (x\right )^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="fricas")

[Out]

1/147840*(49920*sqrt(2)*(cosh(x)^8 + 8*cosh(x)^7*sinh(x) + 28*cosh(x)^6*sinh(x)^2 + 56*cosh(x)^5*sinh(x)^3 + 7
0*cosh(x)^4*sinh(x)^4 + 56*cosh(x)^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8)*sqr
t(a)*weierstrassPInverse(-4, 0, cosh(x) + sinh(x)) + sqrt(2)*(77*cosh(x)^16 + 1232*cosh(x)*sinh(x)^15 + 77*sin
h(x)^16 + 14*(660*cosh(x)^2 + 59)*sinh(x)^14 + 826*cosh(x)^14 + 196*(220*cosh(x)^3 + 59*cosh(x))*sinh(x)^13 +
2*(70070*cosh(x)^4 + 37583*cosh(x)^2 + 2203)*sinh(x)^12 + 4406*cosh(x)^12 + 8*(42042*cosh(x)^5 + 37583*cosh(x)
^3 + 6609*cosh(x))*sinh(x)^11 + 2*(308308*cosh(x)^6 + 413413*cosh(x)^4 + 145398*cosh(x)^2 + 9561)*sinh(x)^10 +
19122*cosh(x)^10 + 4*(220220*cosh(x)^7 + 413413*cosh(x)^5 + 242330*cosh(x)^3 + 47805*cosh(x))*sinh(x)^9 + 6*(
165165*cosh(x)^8 + 413413*cosh(x)^6 + 363495*cosh(x)^4 + 143415*cosh(x)^2)*sinh(x)^8 + 16*(55055*cosh(x)^9 + 1
77177*cosh(x)^7 + 218097*cosh(x)^5 + 143415*cosh(x)^3)*sinh(x)^7 + 2*(308308*cosh(x)^10 + 1240239*cosh(x)^8 +
2035572*cosh(x)^6 + 2007810*cosh(x)^4 - 9561)*sinh(x)^6 - 19122*cosh(x)^6 + 4*(84084*cosh(x)^11 + 413413*cosh(
x)^9 + 872388*cosh(x)^7 + 1204686*cosh(x)^5 - 28683*cosh(x))*sinh(x)^5 + 2*(70070*cosh(x)^12 + 413413*cosh(x)^
10 + 1090485*cosh(x)^8 + 2007810*cosh(x)^6 - 143415*cosh(x)^2 - 2203)*sinh(x)^4 - 4406*cosh(x)^4 + 8*(5390*cos
h(x)^13 + 37583*cosh(x)^11 + 121165*cosh(x)^9 + 286830*cosh(x)^7 - 47805*cosh(x)^3 - 2203*cosh(x))*sinh(x)^3 +
2*(4620*cosh(x)^14 + 37583*cosh(x)^12 + 145398*cosh(x)^10 + 430245*cosh(x)^8 - 143415*cosh(x)^4 - 13218*cosh(
x)^2 - 413)*sinh(x)^2 - 826*cosh(x)^2 + 4*(308*cosh(x)^15 + 2891*cosh(x)^13 + 13218*cosh(x)^11 + 47805*cosh(x)
^9 - 28683*cosh(x)^5 - 4406*cosh(x)^3 - 413*cosh(x))*sinh(x) - 77)*sqrt((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2
*cosh(x)*sinh(x) + sinh(x)^2 + 1)))/(a^3*cosh(x)^8 + 8*a^3*cosh(x)^7*sinh(x) + 28*a^3*cosh(x)^6*sinh(x)^2 + 56
*a^3*cosh(x)^5*sinh(x)^3 + 70*a^3*cosh(x)^4*sinh(x)^4 + 56*a^3*cosh(x)^3*sinh(x)^5 + 28*a^3*cosh(x)^2*sinh(x)^
6 + 8*a^3*cosh(x)*sinh(x)^7 + a^3*sinh(x)^8)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)**3)**(5/2),x)

[Out]

Integral((a*sech(x)**3)**(-5/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*sech(x)^3)^(-5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a/cosh(x)^3)^(5/2),x)

[Out]

int(1/(a/cosh(x)^3)^(5/2), x)

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